Synchronous Gathering without Multiplicity Detection: a Certified Algorithm

Abstract

In mobile robotic swarms, the gathering problem consists in coordinating all the robots so that in finite time they occupy the same location, not known beforehand. Multiplicity detection refers to the ability to detect that more than one robot can occupy a given position. When the robotic swarm operates synchronously, a well-known result by Cohen and Peleg permits to achieve gathering, provided robots are capable of multiplicity detection. We present a new algorithm for synchronous gathering, that does not assume that robots are capable of multiplicity detection, nor make any other extra assumption. Unlike previous approaches, the correctness of our proof is certified in the model where the protocol is defined, using the Coq proof assistant.

Appendix: Axioms of the formalisation

In the main file Gathering/InR2/FSyncFlexNoMultAlgorithm.v, the last command: Print Assumptions Gathering_in_R2 shows all the axioms upon which the proof of correctness of our algorithm for gathering in ${\mathbb {R}}^{2}$ relies, in total 31 axioms. Here, we break them down. They can be classified in three categories:

• The first category is the axiomatisation of reals numbers from the Coq standard library. It represents by far the biggest number of axioms (26), and they are not listed here.

• The second category is the description of the problem.

  

  As one can see, it simply means that our proof is valid for any number nG of robots greater than or equal to 2. Notice that with one robot or less, the problem is not interesting (trivially solved).

• The third category contains three usual geometric properties that are not part of our library. These three axioms are the only ones which could be seen as real axioms to be proved, the previous two categories being the parameters of the problem. On the one hand, there are some properties about barycentres that we think could be provable from its axiomatisation but are currently left as axioms: that the barycentre is unique and that the result of the function computing the barycentre is indeed a barycentre:

  

  On the other hand, there is the proof that similarities can be expressed with an orthogonal matrix M, a zoom factor λ and a translation t: for any similarity s, we can find $M \in \mathcal {O}_{2}(\mathbb {R})$, $\lambda \in \mathbb {R}^{+}$ and $t \in \mathbb {R}^{2}$ such that s = λM + t. For convenience, the orthogonal matrix and the zoom factor are combined into two column vectors u and v: we have λM = (u v) with u ⊥ v and ∥u∥ = ∥v∥ = λ.

  

  These types of axioms can be discharged through the connection with Coq libraries dedicated to geometry.